$H_{\infty}$ Control of Discrete-Time Systems With Multiple Input Delays

This paper is concerned with the finite horizon H
_infin
full-information control for discrete-time systems with multiple control and exogenous input delays. We first establish a duality between the H
_infin
full-information control and the H
₂
smoothing of a stochastic backward system in Krein space. Like the duality between the linear quadratic regulation (LQR) of linear systems without delays and the Kalman filtering, the established duality allows us to address complicated multiple input delay problems in a simple way. Indeed, by applying innovation analysis and standard projection in Krein space, in this paper we derive conditions under which the H
_infin
full-information control is solvable. An explicit controller is constructed in terms of two standard Riccati difference equations of the same order as the original plant (ignoring the delays). As special cases, solutions to the H
_infin
state feedback control problem for systems with delays only in control inputs and the H
_infin
control with preview are obtained. An example is given to demonstrate the effectiveness of the proposed H
_infin
control design